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Math Help - Derivative Word Problem Help Please!~

  1. #1
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    Derivative Word Problem Help Please!~

    An observatory is to be in the form of a right circular cylinder surmounted by a hemispherical dome. If the hemispherical dome costs 6 times as much per square foot as the cylindrical wall, what are the most economic dimensions for a volume of 16000 cubic feet?

    The radius of the cylindrical base (and of the hemisphere) is _____ ft. (Round to the nearest tenth).
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    Re: Derivative Word Problem Help Please!~

    Here are some questions to help guide you. What are the equations of surface area for the lateral surface area of a cylinder, and that of a hemisphere? What would be the cost of the cylindrical part if the cost was C per ft^2 (if you need more help: you want to buy 2 kg of apples and the price is $3 per kg: what do you do to these two numbers to figure out the total cost?) Post your answers and further questions below and we'll go from there.
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    Re: Derivative Word Problem Help Please!~

    Quote Originally Posted by majamin View Post
    Here are some questions to help guide you. What are the equations of surface area for the lateral surface area of a cylinder, and that of a hemisphere? What would be the cost of the cylindrical part if the cost was C per ft^2 (if you need more help: you want to buy 2 kg of apples and the price is $3 per kg: what do you do to these two numbers to figure out the total cost?) Post your answers and further questions below and we'll go from there.
    π = pi

    Area: A = 2πr2+2πrh
    C = (6)2πr2+2πrh

    Cost: C = (32000/r) + (72πr2/3)

    This could be wrong.. I'm not very good with these word problems. Are these equations correct?
    Last edited by michellederz; March 9th 2013 at 01:25 PM.
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    Re: Derivative Word Problem Help Please!~

    Quote Originally Posted by michellederz View Post
    π = pi

    Area: A = 2πr2+2πrh
    C = (6)2πr2+2πrh
    Good start. Because we are just calculating the wall of a cylinder the equation has to change to A=2\pi rh (it doesn't include 2\pi r^2). The area of a hemisphere is half of 4\pi r^2 or 2\pi r^2. Let the cost of the cylindrical portion be C, so the hemispherical portion is 6C.

    The cost will be Cost = (CylindricalPortion) \times C + (HemisphericalPortion) \times 6C

    Cost: C = (32000/r) + (72πr2/3)

    This could be wrong.. I'm not very good with these word problems. Are these equations correct?
    . I get Cost = C\left(\frac{32000}{r}+\frac{32}{3}\pi r^2\right) where C is the cost as above. We need to minimize cost, so this is an opportune time for the derivative (I'll let you try to fill in the details). As a final answer, I get r=7.8 ft.
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    Re: Derivative Word Problem Help Please!~

    Okay I think I got it now! Thank you so very much!
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